Hanlon, P. Invent Math (1986) 86: 131. doi:10.1007/BF01391498
LetA+(k) denote the ring ℂ[t]/tk+1 and letG be a reductive complex Lie algebra with exponentsm1, ...,m n. This paper concerns the Lie algebra cohomology ofG⊗A+(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofG⊗A+(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2ms+1 fors=1,2, ...,n. Of thesek+1 generators of degree 2ms+1, one has weight 0 and the others have weights (k+1)ms+t fort=1,2, ...,k.
It is shown that this conjecture about the Lie algebra cohomology of ⊗A+(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemAn,Bn,Cn, orDn. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.